162 INDUCTION. 



respect than position or magnitude, as well as inde- 

 pendently of the physical cause from which in any 

 particular case they happen to derive their origin. 



It thus appears that mathematics is the only de- 

 partment of science into the methods of which it still 

 remains to inquire. And there is the less necessity 

 that this inquiry should occupy us long, as we have 

 already, in the second book, made considerable pro- 

 gress in it. We there remarked, that the directly 

 inductive truths of mathematics are few in number ; 

 consisting of the axioms, together with certain propo- 

 sitions concerning existence, tacitly involved in most 

 of the so-called definitions. And we proved, at such 

 length as makes any return to the subject altogether 

 superfluous, that these original premisses,, from which 

 the remaining truths of the science are deduced, 

 are, notwithstanding all appearances to the contrary, 

 results of observation and experience; founded, in 

 short, on the evidence of the senses. That things 

 equal to the same thing are equal to another, or that 

 two straight lines which have once intersected with 

 one another continue to diverge, are inductive truths ; 

 resting indeed, like the law of universal causation, 

 only upon induction per enumerationem simplicem; 

 upon the fact that they have been perpetually found 

 true and never once false. But as we have seen in a 

 recent chapter that this evidence, in the case of a law 

 so completely universal as the law of causation, 

 amounts to the fullest proof attainable by the human 

 faculties, so is this even more evidently true of the 

 general propositions to which we are now adverting : 

 because, as a perception of their truth in any individual 

 case whatever, requires only the simple act of looking at 

 the objects in a proper position, there never could have 

 been in their case (what, for a long period, in the case 



